Tuesday, February 11, 2014

Shuffling

Pile shuffling isn't shuffling.

I've heard this a lot, and it came up recently in a discussion thread I was reading. Randomization and fairness are important elements to any card game, including Magic. The goal of shuffling is twofold:

1) Every order of cards is equally likely.
2) Neither player has information about the order of the cards that they shouldn't.

The second goal follows somewhat from the first, but not entirely. For example, a player's shuffling technique could result in a properly randomized order of cards, but might have revealed information during the actual shuffling (e.g. by showing the bottom card to the shuffler during the process of shuffling).

To determine whether a given method for shuffling is randomizing well, we would need to generate some sample of randomized cards using the technique (the larger the sample the better), and analyze whether or not there is a systematic bias. That means, are certain orders of cards occurring more often than they should be. If so, the shuffling method is not doing a good job. How well the method is performing is dependent on how systematic the bias is.

Riffle shuffling consists of dividing a deck of cards into two piles and interleaving them back into one pile

Now then:

Wait, what? The extent to which a riffle shuffle meets the goals posted above, then a single riffle shuffle is a valid part of a good shuffling method. However, a highly-skilled shuffler can riffle shuffle such that they interleave the cards perfectly, giving them exact information about the location of the cards. If a player is skilled at deck manipulation, then riffle shuffling is just as invalid as pile shuffling.

But what about pile shuffling? Some say "it's just counting, not shuffling". Well, that's not true if, as part of your entire shuffling routine the pile shuffle helps in meeting the goals above. It's true that if you pile shuffle in exactly the same way with the right number of repetitions, you will return to the same exact configuration. That doesn't mean that a single pile shuffle is not a valid part of an entire shuffling method, though. That means that a certain number of pile shuffles with no other actions is a very bad shuffling technique. Well, the same holds for riffle shuffling, if done in a particular way.

If a player pile shuffles once, then riffle shuffles several times, then pile shuffles again. Assuming they do not know the starting configuration of the deck and assuming they are not a skilled deck manipulator, then the pile shuffle will actually contribute to the overall shuffling method. A single pile shuffle insures that adjacent cards are non-adjacent. If the player knew that before shuffling the top two cards were islands, a pile shuffle would mean the player no longer knew this information (though they would know that the islands are x cards apart, where x is the number of piles). But this information is quite a bit more difficult to hold in the working memory of your average player. In general, a pile shuffle is going to make it more difficult for a player to remember the locations of individual cards, even if they knew the starting configuration.

In Magic, the pile shuffle has the added advantage that it does work as a count, making sure that you have the right number of cards in your deck. But it gets an unfair rap as "not real shuffling", probably because if done without any other actions or in a certain way it can lead to systematic bias and potentially give a player information about the configuration of the cards. But the same is true of riffle shuffling.

So stop your hating. Pile shuffling is a valid part of a good shuffling technique. You just don't pile shuffle exclusively and you'll be fine. I'm sure we could validate this premise with either experiments or simulation. Supposedly experiments have demonstrated that seven riffle shuffles is the magic number for good randomization of a standard 52-card deck. My guess is that replacing some of those riffle shuffles with pile shuffles would yield good results as well.